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A374521
Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of anti-runs sum to k.
6
1, 0, 1, 0, 0, 2, 0, 1, 1, 2, 0, 2, 1, 2, 3, 0, 2, 5, 3, 4, 2, 0, 5, 7, 8, 3, 5, 4, 0, 9, 12, 11, 17, 5, 8, 2, 0, 14, 26, 23, 22, 24, 6, 9, 4, 0, 25, 42, 54, 41, 36, 36, 7, 12, 3, 0, 46, 76, 88, 107, 60, 60, 48, 9, 14, 4
OFFSET
0,6
COMMENTS
The leaders of anti-runs in a sequence are obtained by splitting it into maximal consecutive anti-runs (sequences with no adjacent equal terms) and taking the first term of each.
EXAMPLE
Triangle begins:
1
0 1
0 0 2
0 1 1 2
0 2 1 2 3
0 2 5 3 4 2
0 5 7 8 3 5 4
0 9 12 11 17 5 8 2
0 14 26 23 22 24 6 9 4
0 25 42 54 41 36 36 7 12 3
0 46 76 88 107 60 60 48 9 14 4
0 78 144 166 179 176 101 83 68 10 17 2
0 136 258 327 339 311 299 139 122 81 12 18 6
0 242 457 602 704 591 544 447 198 165 109 12 23 2
Row n = 6 counts the following compositions:
. (15) (24) (321) (42) (51) (6)
(141) (114) (312) (1122) (411) (33)
(132) (231) (1113) (11112) (3111) (222)
(123) (213) (2112) (2211) (111111)
(1212) (1311) (1221) (21111)
(1131) (12111)
(2121) (11211)
(11121)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, UnsameQ]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Column n = k is A000005, except a(0) = 1.
Row-sums are A011782.
Column k = 1 is A096569.
For length instead of sum we have A106356.
The corresponding rank statistic is A374516, row-sums of A374515.
For identical leaders we have A374517, ranks A374519.
For distinct leaders we have A374518, ranks A374638.
Other types of runs (instead of anti-):
- For leaders of identical runs we have A373949.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
Sequence in context: A342652 A083715 A037135 * A202177 A354793 A029360
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Aug 02 2024
STATUS
approved